VTRMC 2015 Problem 7

Let $n$ be a positive integer and let $x_1,\ldots,x_n$ be $n$ nonzero points in $\mathbb R^2$ . Suppose $\langle x_i,x_j\rangle$ (scalar or dot product) is a rational number for all $i,j$ ( $1\le i,j\le n$ ). Let $S$ denote all points of $\mathbb R^2$ of the form $\sum_{i=1}^na_ix_i$ where the $a_i$ are integers. A closed disk of radius $R$ and center $P$ is the set of points at distance at most $R$ from $P$ (includes the points distance $R$ from $P$ ). Prove that there exists a positive number $R$ and closed disks $D_1,D_2,\ldots$ of radius $R$ such that

(a) Each disk contains exactly two points of $S$ ; (b) Every point of $S$ lies in at least one disk; (c) Two distinct disks intersect in at most one point.